Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators

Jian Wang; Meng-ge Zhang; Takashi Kumagai

REVIEW 2 major objections 3 minor 1 cited by

Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Subsolutions of fractional p-Laplacian parabolic equations are locally bounded for every p>1, with a bound that matches the linear case p=2.

desk verdict Real improvement in local boundedness for fractional p-Laplacian parabolic equations, but the main theorem is stated too loosely: the proof needs extra integrability that isn't in the hypotheses. read the letter →

arxiv 2412.03770 v1 pith:NUIAMOL5 submitted 2024-12-04 math.AP

Takashi Kumagai , Jian Wang , Meng-ge Zhang This is my paper · ORCID

classification math.AP MSC 35B6535R1135K5560J75

keywords fractionalp-LaplacianlocalboundednessparabolicequationsDeGiorgi-Nash-MoseriterationCaccioppoliinequalitynonlocaltailsubsolutionSobolevspaces

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that every local subsolution of a parabolic equation driven by a fractional p-Laplacian type operator is locally bounded above, for every p in (1,∞). This fills a gap: prior results either handled only p≥2, or used a parabolic tail with an L∞-norm in time that never degenerates to the linear p=2 estimate. The new bound is explicit, with the local supremum controlled by local mean values of the positive part and a single nonlocal parabolic tail that uses only the L1-norm in time. A reader should care because this is the missing boundedness step needed to build Hölder regularity and Harnack estimates for these nonlinear nonlocal parabolic equations.

What carries the argument

The proof is carried by a Caccioppoli-type inequality (Lemma 2.1) for the truncated function v=(u−k)_+ raised to a power w=$v^{{(p−1+ξ)/p}}$, together with a decomposition of the nonlocal supremum tail into an intermediate tail, controlled by the equation, and a remaining tail, controlled by cancellation inside the iteration. A De Giorgi–Nash–Moser iteration then runs in two regimes, p≥2 and p∈(1,2), using fractional Sobolev inequalities with different exponents.

What would settle it

Take p≥2 and construct a function u on a cylinder satisfying the weak subsolution inequality (1.3) with finite local means of u_+^{2p−2}, u_+^p, and finite tail Tail_{p−1}, but with sup of u over B_{R/2}×I_{R/2}^⊖ infinite. Theorem 1.3 would be false. A concrete candidate would be a solution with a growing singularity near the boundary of the smaller cylinder; numerical or analytic construction of such an example would settle the claim.

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Extended reading notes

Core claim

Theorem 1.3 is the central claim: if u satisfies ∂_t u − L_t u ≤ 0 on a backward cylinder, then sup over the half-size cylinder is at most constants times sums of local means of powers of u_+ (2p−2 and p for p≥2; 2p−2, p, and 1+ξ for p∈(1,2)) plus the parabolic tail Tail_{p−1}, defined with the L1-norm in time. The statement is specifically designed so that when p=2 it reproduces the known linear estimate exactly, without the extra constant term that appears in earlier fractional p-Laplacian results.

Load-bearing premise

The estimate is meaningful only if the right-hand side quantities are finite: u_+ must lie in the stated Lebesgue classes ($L^{{2p−2}}$ for p≥2, $L^{{1+ξ}}$ with ξ>d(2−p)/(sp) for p<2) and the parabolic tail must be finite; the theorem's stated hypothesis, the pointwise inequality (1.3), does not by itself guarantee this.

Editorial extensions

If this is right

If u is a subsolution to ∂_t u − L_t u = f with f ∈ L^1_{t,x}, then the transformation v(t,x)=u(t,x)−∫_{t0}^t ‖f(s,·)‖_{L∞(B_R)} ds reduces to the homogeneous case, so boundedness transfers to equations with source terms.
For p=2 the estimate collapses to the linear bound with (∫∫ u_+^2)^{1/2} plus Tail_1, matching the known linear parabolic result.
For p∈(1,2), choosing the allowed ξ yields an explicit finite bound, so local weak subsolutions in the stated integrability class are locally bounded for every p>1, including the previously restricted range.
The L1-in-time parabolic tail replaces the L∞-in-time tail used in earlier works, which is the natural weakening needed to align nonlinear theory with the linear case.
The paper positions boundedness as a basis for later Hölder continuity and Harnack estimates for these equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Not in the paper: a natural next step is to use Theorem 1.3 in a Moser or Krylov–Safonov iteration to obtain Hölder continuity for all p>1; the linear case p=2 already has such a theory, and the boundedness proved here is the missing input.
Not in the paper: the time-dependent level k(t) chosen to absorb the nonlocal tail suggests the same truncation could handle kernels comparable to |x−y|^{−d−sp} with slowly varying coefficients, such as variable-order s(x,y), where an L1-tail would be harder to define.
Not in the paper: one could test whether the exponent 2p−2 in the p≥2 bound is necessary by constructing extremal functions where the u_+^p term is small and the supremum is controlled by the tail; the paper does not address optimality.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper does what it claims, but the headline theorem overclaims. The De Giorgi iteration is coherent and the new Caccioppoli inequality with the free parameter ξ is a genuine advance, letting the tail use only L1-in-time for all p>1. That matches the linear p=2 theory and improves [1], [9], [15]. If the paper were just a refinement of known tools, I'd call it routine; the level-set truncation in Prop 2.6 that cancels the intermediate tail is the clever part, and it seems to work.

Soft spots in order of importance. First, Theorem 1.3 states only that u is a subsolution in the pointwise sense. The proof of the p≥2 case applies Prop 2.5 to u_+ and needs u_+ in L^{2p−2} locally; the p<2 case needs u_+ in L^{1+ξ} with ξ>d(2−p)/(sp). Those are not consequences of the natural energy space L^p(I;W^{s,p})∩L∞(I;L^2). So as stated, the theorem is a conditional estimate: if the displayed right-hand side is infinite, it says nothing, and the paper does not show these norms are finite for the class it claims to cover. This is fixable — state the theorem with explicit integrability hypotheses, or add a preliminary step proving them under the energy-class assumption. But as written it is a real gap in the statement, not just a cosmetic one.

Second, Prop 2.5, Case 2 has a typo: the coefficient p/(p−1) in the iteration inequality should be (p−1)/p. Otherwise κ>1 and the iteration doesn't close. A referee will catch it, but it should be fixed. Third, minor: Lemma 2.2 is applied to f(r)=sup_t ∮ v^p without proving f bounded; the standard truncation argument is omitted. Also two references ([2] and [8]) are never cited.

I did not re-derive the Sobolev embeddings borrowed from [9], but they look standard and the iteration algebra checks out. The core method is sound.

Who this is for: people working on nonlocal parabolic regularity. It deserves a serious referee; the contribution is real even though the presentation needs work. My recommendation: send to peer review with a request to tighten the statement and fix the typo.

Referee Report

2 major / 3 minor

Summary. The paper proves a quantitative local boundedness estimate for subsolutions of the parabolic equation ∂_t u − L_t u = 0, where L_t is a fractional p-Laplacian type operator with kernel comparable to |x−y|^{−d−sp}, for every p∈(1,∞). The estimate controls sup_{B_{R/2}×I_{R/2}^⊖} u by local averages of powers of u_+ and by a nonlocal parabolic tail with L^1-in-time weight. The proof combines a new Caccioppoli-type inequality (Lemma 2.1), a tail-cancellation mechanism (Proposition 2.6), and a De Giorgi iteration with carefully chosen level sets. The paper claims to extend the linear p=2 result of Kassmann and Weidner to all p>1.

Significance. If the estimate holds for the intended weak-solution class, this is a valuable contribution: it unifies the p=2 linear theory with the nonlinear fractional p-Laplacian case and improves earlier results by using an L^1-in-time tail rather than an L^∞-in-time tail. The proof is coherent and genuinely structural: the constants are explicit and no parameter is fitted, and Proposition 2.6's cancellation of the outer tail is an elegant and checkable step. The main deficit is that the theorem statement does not specify the integrability and regularity hypotheses that the proof actually uses, so the advertised local boundedness result is only conditional as stated.

major comments (2)

[§1.2, Theorem 1.3; §3, (3.3) and (3.4)] The theorem as stated is broader than what the proof establishes. The proof applies Proposition 2.5 to u_+ at (3.3), which requires u_+ ∈ L^{2p−2}(I_R^⊖×B_R) when p≥2. For p∈(1,2), the iteration starts from A_0 = ∬ u_+^{1+ξ} with ξ > d(2−p)/(sp), and Proposition 2.6 and the subsequent estimates require this integral and Tail_{p−1}(u) to be finite. None of these conditions is stated in Theorem 1.3, and none follows from the pointwise inequality (1.3) or from the natural energy class L^p(I;W^{s,p})∩L^∞(I;L^2). For example, when p=4, d=1, s=0.1, Lemma 2.4(i) gives at most u_+∈L^{4.8} from the energy class, whereas 2p−2=6 is needed. If the displayed averages are infinite, the estimate is vacuous and no boundedness follows. To make Theorem 1.3 a local boundedness statement for weak subsolutions, the authors must either prove these integrability properties for their solution class or state them explicitly as hypotheses; they should also specify the global spatial class needed to define the tail.
[§2, Proposition 2.5, Case 2 (sp≤1)] In the proof of Proposition 2.5, after the case sp≤1 is treated, the displayed iteration inequality reads f(r) ≤ (s−r)^{−1}(A_1+A_2) + (s−r)^{−p(d+sp)}B + [p/(p−1)] f((r+s)/2). Lemma 2.2 is then invoked, but it requires a contraction coefficient κ<1, whereas p/(p−1)>1 for p>1. The preceding calculation gives (p−1)/p as that coefficient, so this appears to be a typographical error; if the displayed coefficient is literal, the iteration argument for sp≤1 is invalid. The authors should correct the coefficient and re-verify that Lemma 2.2 applies in both cases.

minor comments (3)

[§2, Proposition 2.5] Lemma 2.2 is applied to f(r)=sup_{I_r^⊖} ∮_{B_r} v(t,x)^p dx without first proving that f is bounded on [R/2,R]. The standard truncation f_N=min(f,N), followed by N→∞, supplies the missing justification and should be mentioned.
[§1.1.1 and §2, Lemma 2.1] The paper defines subsolutions through the pointwise inequality (1.3) but all later calculations, especially the integration by parts in time in Lemma 2.1, are formal. The authors should state the weak/energy formulation of (1.1) and the class of admissible test functions for which Lemma 2.1 is proved.
[Throughout] There are several typographical slips: 'supsolution' in §1.1.1, 'For the safe of completeness' at the start of Part (2) of the proof of Theorem 1.3, and inconsistent notation such as 'p−1/p' where '(p−1)/p' is meant. These should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorem is a genuine a priori estimate with structural constants, and the cited external lemmas are independent support.

full rationale

The paper derives Theorem 1.3 through a direct De Giorgi–Nash–Moser iteration based on Lemma 2.1 (Caccioppoli), Proposition 2.5, and Proposition 2.6. The Caccioppoli inequality is proved from the subsolution inequality (1.3) by an explicit test-function computation; the constants depend only on p, ξ, d, s, and the kernel comparability constant. Proposition 2.6 chooses C := c2c6 to cancel an upper tail bound against a negative term from the equation; this is a legitimate absorption of a term by choosing a constant, not a circular definition. The final bound is an a priori estimate: the right-hand side consists of local Lp, L2p−2, or L1+ξ averages and the nonlocal parabolic tail, none of which is fitted to the target sup. The paper cites external results (e.g., Lemma 1.1 from Strömqvist [15], Lemma 2.4 from Ding–Zhang–Zhou [9]) for standard tools; these are not authored by the present paper and are not load-bearing in a circular sense. The self-citations [3,4] appear only in the literature overview, not in the proof. The main caveat, noted by the reader, is that Theorem 1.3 is stated only under the pointwise subsolution condition (1.3) while the proof requires finiteness of the displayed norms, such as u_+ in L^{2p−2} for p ≥ 2 or L^{1+ξ} for p ∈ (1,2), and of the tail. That is a hypothesis/regularity gap that affects the non-vacuous interpretation of the theorem, but it is not a circular reduction: the theorem would simply need these finiteness assumptions stated. No quantity used in the proof is equivalent to the conclusion by definition, and no fitted parameter is renamed as a prediction. The derivation chain is self-contained once the stated integrability hypotheses are imposed.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data; constants depend only on p, d, s, Λ. The central estimate relies on the standard kernel comparability (1.2), the subsolution property of the positive part ([15, Lemma 3.1]), fractional Sobolev embeddings from [9], and iteration lemmas from [10], [11]. The main hidden input is an integrability condition on u that is not stated in Theorem 1.3.

assumptions (6)

domain assumption Kernel comparability (1.2): Λ^{-1}|x−y|^{−d−sp} ≤ J(t;x,y) ≤ Λ |x−y|^{−d−sp} and symmetry J(t;x,y)=J(t;y,x)
Defines the class of nonlocal operators studied; used throughout, e.g., in Lemma 2.1 to bound J in the Caccioppoli inequality.
domain assumption The positive part u_+ of a subsolution is again a subsolution (Lemma 1.1, cited from [15, Lemma 3.1])
Used to reduce the theorem to nonnegative subsolutions and to apply Proposition 2.5 to u_+.
standard math Parabolic fractional Sobolev embeddings (Lemma 2.4, cited from [9, Lemmas 2.3, 2.4])
Used in Step (i) of both cases of Theorem 1.3 to control space-time L^q norms by the Gagliardo seminorm and a sup-in-time L^2 or L^p norm.
standard math Giaquinta-Giusti iteration lemma (Lemma 2.2, cited from [10, Lemma 1.1])
Used in Proposition 2.5 to absorb the sup-in-time term and get the L^p estimate.
standard math Elementary iteration lemma (Lemma 2.3, see [11, Lemma 7.1])
Used to conclude convergence of the De Giorgi iteration levels A_i to 0.
ad hoc to paper Implicit integrability of u: the estimates require u_+ ∈ L^{2p−2}(I_R^⊖×B_R) for p≥2 and u_+ ∈ L^{1+ξ} for p∈(1,2), plus finiteness of the parabolic tail; the theorem statement does not make these explicit.
The Caccioppoli proof in Lemma 2.1 and the tail definition (1.4) require these integrability conditions; without them the RHS of Theorem 1.3 may be infinite and the statement is vacuous or formal.

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Pith. "Pith review of Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators." pith.science (2026). https://pith.science/paper/NUIAMOL5

@misc{pith2026241203770,
  author       = {Pith},
  title        = {Pith review of: Local boundedness of solutions to parabolic equations associated with fractional $p$-Laplacian type operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NUIAMOL5}},
  note         = {Machine review of arXiv:2412.03770}
}

abstract

In this paper, we study the local boundedness of local weak solutions to the following parabolic equation associated with fractional $p$-Laplacian type operators $$ \partial_t u(t,x)-\text{p.v.}\int_{\R^d}|u(t,y)-u(t,x)|^{p-2}(u(t,y)-u(t,x))J(t;x,y)\,dy=0,\quad (t,x)\in \R\times \R^d, $$ where $\text{p.v.}$ means the integral in the principal value sense, $p\in(1,\infty)$ and $J(t;x,y)$ is comparable to the kernel of the fractional $p$-Laplacian operator $|x-y|^{-d-sp}$ with $s\in(0,1)$ and uniformly in $(t;x,y)\in\R\times\R^d\times\R^d$. Unlike existing results in the literature, the local boundedness of the solutions obtained in this paper extends the known results for the linear case (i.e., the case that $p=2$), in particular with a nonlocal parabolic tail that uses the $L^1$-norm in time for all $p\in (1,\infty)$. The proof is based on a new level set truncation in the De Giorgi-Nash-Moser iteration and a careful choice of iteration orders, as well as a general Caccioppoli-type inequality that is efficiently applied to fractional $p$-Laplacian type operators with all $p>1$.

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Works this paper leans on

17 extracted references · 16 canonical work pages · cited by 1 Pith paper

[1]

and Kim, K.: A Hölder estimate with an optimal t ail for nonlocal parabolic p-Laplace equations, Annali di Matematica, 203 (2024), 109–147

Byun, S.S. and Kim, K.: A Hölder estimate with an optimal t ail for nonlocal parabolic p-Laplace equations, Annali di Matematica, 203 (2024), 109–147

work page 2024
[9]

and Zhou, S.L.: Local boundedness a nd Hölder continuity for the parabolic fractional p-Laplace equations, Calc

Ding, M.Y., Zhang, C. and Zhou, S.L.: Local boundedness a nd Hölder continuity for the parabolic fractional p-Laplace equations, Calc. Var., 60 (2021), Paper no. 38

work page 2021
[15]

Diﬀerential Equations , 266 (2019), 7948–7979

Strömqvist, M.: Local boundedness of solutions to non- local parabolic equations modeled on the fractional p-Laplacian, J. Diﬀerential Equations , 266 (2019), 7948–7979

work page 2019
[2]

and Strömqvist, M.: Continuity of solutions to a nonlinear fractional diﬀusion equation, J

Brasco, L., Lindgren, E. and Strömqvist, M.: Continuity of solutions to a nonlinear fractional diﬀusion equation, J. Diﬀerential Equations , 21 (2021), 4319–4381

work page 2021
[8]

and Valdinoci, E.: Hitchhike r’s guide to the fractional Sobolev spaces, Bull

Di Nezza, E., Palatucci, G. and Valdinoci, E.: Hitchhike r’s guide to the fractional Sobolev spaces, Bull. Sci. Math. , 136 (2012), 521–573

work page 2012
[3]

and Wang, J.: Elliptic Harnack i nequalities for symmetric non-local Dirichlet forms, J

Chen, Z.-Q., Kumagai, T. and Wang, J.: Elliptic Harnack i nequalities for symmetric non-local Dirichlet forms, J. Math. Pures Appl. , 125 (2019), 1–42

work page 2019
[4]

and Wang, J.: Stability of parab olic Harnack inequalitiess for symmetric non-local Dirich let forms, J

Chen, Z.-Q., Kumagai, T. and Wang, J.: Stability of parab olic Harnack inequalitiess for symmetric non-local Dirich let forms, J. Eur. Math. Soc. , 22 (2020), 3747–3803

work page 2020
[5]

Cozzi, M.: Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a uniﬁed approach via fractional De Giorgi classes, J. Funct. Anal. , 272 (2017), 4762–4837

work page 2017

Show all 17 references

[6]

and Palatucci, G.: Local behavio r of fractional p-minimizers, Ann

Di Castro, A., Kuusi, T. and Palatucci, G.: Local behavio r of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279–1299

2016
[7]

and Palatucci, G.: Nonlocal Harn ack inequalities, J

Di Castro, A., Kuusi, T. and Palatucci, G.: Nonlocal Harn ack inequalities, J. Funct. Anal. , 267 (2014), 1807–1836

2014
[10]

and Giusti, E.: On the regularity of the mi nima of variational integrals, Acta Math., 148 (1982), 31–46

Giaquinta, M. and Giusti, E.: On the regularity of the mi nima of variational integrals, Acta Math., 148 (1982), 31–46

1982
[11]

Giusti, E.: Direct Methods in the Calculus of Variations , World Scientiﬁc, 2003

2003
[12]

and Weidner, M.: The parabolic Harnack ine quality for nonlocal equations, to appear in Duke J

Kassmann, M. and Weidner, M.: The parabolic Harnack ine quality for nonlocal equations, to appear in Duke J. Math. , see arXiv: 2303.05975

arXiv
[13]

and Toledo, J.: Fractional p-L aplacian evolution equations, J

Mazón, J.M., Rossi, J.D. and Toledo, J.: Fractional p-L aplacian evolution equations, J. Math. Pures Appl. , 105 (2016), 810–844. LOCAL BOUNDEDNESS OF PARABOLIC FRACTIONAL p-LAPLACIAN TYPE EQUATIONS 25

2016
[14]

and Tewary, V.: Local boundedness of variati onal solutions to nonlocal double phase parabolic equation s, J

Prasad, H. and Tewary, V.: Local boundedness of variati onal solutions to nonlocal double phase parabolic equation s, J. Diﬀerential Equations , 351 (2023), 243–276

2023
[16]

Strömqvist, M.: Harnack’s inequality for parabolic no nlocal equations, Ann. Inst. H. Poincaré Anal. Non Linéaire , 36 (2019), 1709–1745

2019
[17]

Diﬀerential Equations , 260 (2016), 6038–6056

Vázquez, J.L.: The Dirichlet problem for the fractiona l p-Laplacian evolution equation, J. Diﬀerential Equations , 260 (2016), 6038–6056. T akashi Kumagai: Department of Mathematics, Waseda University, Shinjuku, T okyo, 169-8555, Japan. t-kumagai@waseda.jp Jian W ang: School ...

2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

and Kim, K.: A Hölder estimate with an optimal t ail for nonlocal parabolic p-Laplace equations, Annali di Matematica, 203 (2024), 109–147

Byun, S.S. and Kim, K.: A Hölder estimate with an optimal t ail for nonlocal parabolic p-Laplace equations, Annali di Matematica, 203 (2024), 109–147

work page 2024

[2] [9]

and Zhou, S.L.: Local boundedness a nd Hölder continuity for the parabolic fractional p-Laplace equations, Calc

Ding, M.Y., Zhang, C. and Zhou, S.L.: Local boundedness a nd Hölder continuity for the parabolic fractional p-Laplace equations, Calc. Var., 60 (2021), Paper no. 38

work page 2021

[3] [15]

Diﬀerential Equations , 266 (2019), 7948–7979

Strömqvist, M.: Local boundedness of solutions to non- local parabolic equations modeled on the fractional p-Laplacian, J. Diﬀerential Equations , 266 (2019), 7948–7979

work page 2019

[4] [2]

and Strömqvist, M.: Continuity of solutions to a nonlinear fractional diﬀusion equation, J

Brasco, L., Lindgren, E. and Strömqvist, M.: Continuity of solutions to a nonlinear fractional diﬀusion equation, J. Diﬀerential Equations , 21 (2021), 4319–4381

work page 2021

[5] [8]

and Valdinoci, E.: Hitchhike r’s guide to the fractional Sobolev spaces, Bull

Di Nezza, E., Palatucci, G. and Valdinoci, E.: Hitchhike r’s guide to the fractional Sobolev spaces, Bull. Sci. Math. , 136 (2012), 521–573

work page 2012

[6] [3]

and Wang, J.: Elliptic Harnack i nequalities for symmetric non-local Dirichlet forms, J

Chen, Z.-Q., Kumagai, T. and Wang, J.: Elliptic Harnack i nequalities for symmetric non-local Dirichlet forms, J. Math. Pures Appl. , 125 (2019), 1–42

work page 2019

[7] [4]

and Wang, J.: Stability of parab olic Harnack inequalitiess for symmetric non-local Dirich let forms, J

Chen, Z.-Q., Kumagai, T. and Wang, J.: Stability of parab olic Harnack inequalitiess for symmetric non-local Dirich let forms, J. Eur. Math. Soc. , 22 (2020), 3747–3803

work page 2020

[8] [5]

Cozzi, M.: Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a uniﬁed approach via fractional De Giorgi classes, J. Funct. Anal. , 272 (2017), 4762–4837

work page 2017